Equilibrium sampling for biomolecules under mechanical tension

Abstract

In the studies of force-induced conformational transitions of biomolecules, the large timescale difference from experiments presents the challenge of obtaining convergent sampling for molecular dynamics simulations. To circumvent this fundamental problem, an approach combining the replica-exchange method and umbrella sampling (REM-US) was developed to simulate mechanical stretching of biomolecules under equilibrium conditions. Equilibrium properties of conformational transitions can be obtained directly from simulations without further assumptions. To test the performance, we carried out REM-US simulations of atomic force microscope (AFM) stretching and relaxing measurements on the polysaccharide pustulan, a (1-->6)-beta-D-glucan, which undergoes well-characterized rotameric transitions in the backbone bonds. With significantly enhanced sampling convergence and efficiency, the REM-US approach closely reproduced the equilibrium force-extension curves measured in AFM experiments. Consistent with the reversibility in the AFM measurements, the new approach generated identical force-extension curves in both stretching and relaxing simulations-an outcome not reported in previous studies, proving that equilibrium conditions were achieved in the simulations. REM-US may provide a robust approach to modeling of mechanical stretching on polysaccharides and even nucleic acids.

Figure 1

Comparison of force spectra obtained from the REM-US simulations (black solid line) and AFM experiments (blue dash-dot lines) (43). Three major conformations involved in the stretching process are shown in the plot, corresponding to the rotamers: gauche-gauche (gg), with the dihedral angle O₆-C₆-C₅-O₅ (denoted by the red dashed lines in the molecules), ω = −60°; gauche-trans (gt), with ω = +60°; and trans-gauche (tg), with ω = 180°.

Figure 2

Force spectra of the stretching process obtained using SMD (blue dash-dot line), umbrella sampling (magenta dash line), and REM-US (red solid line). (Inset) Zoom-in of the region 5.5–6.0 Å, to better demonstrate the discrepancies of the force among different methods.

Figure 3

Convergence comparison of the force spectra for the stretching, relaxing, and parallel processes using the umbrella sampling and the REM-US approach. (Inset) Zoom-in of the region 5.5–6.0 Å.

Figure 4

Trajectories of dihedral angles ω (black), t1 (cyan), and t2 (magenta) from the REM-US simulation. The dihedral angle ω characterizes the conformational changes from rotamer gg and gt to tg.

Figure 5

Time evolution of the dihedral angle ω of ring 5 in the trajectories with x_i = 58.0 Å. (Black trace) Umbrella sampling simulation; (red trace) REM-US simulation. The wide distribution of the dihedral angles ω from the REM-US simulation indicates the frequent conformational changes among different rotamers gg, gt, and tg.